Theorem fusgrvtxfi 29298

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29297	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 496	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  Fincfn 8959  Vtxcvtx 28975  USGraphcusgr 29128  FinUSGraphcfusgr 29295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-fusgr 29296

This theorem is referenced by:  fusgrfupgrfs  29310  nbfusgrlevtxm1  29356  nbfusgrlevtxm2  29357  nbusgrvtxm1  29358  uvtxnm1nbgr  29383  cusgrm1rusgr  29562  wlksnfi  29889  fusgrhashclwwlkn  30060  clwwlkndivn  30061  fusgreghash2wsp  30319  numclwwlk3lem2  30365  numclwwlk4  30367